pt on ground 70 ft from the base of a tree distance to top of tree is 2 ft more than 3 times the height. Find the height of the tree.

Question

Quadratic applications!

Answer 1

I assume you mean the distance to the top of the tree is from the point 70 feet from the base of the tree.

This forms a right triangle whereby one leg is 70 feet and the height, which is unknown, is the other.

The hypotenuse is the diagonal distance from the point to the top of the tree.

Let h=height of the tree

Using the Pythagorean Theorem...

(3h+2)²=70²+h²

9h²+12h+4=4900+h²

9h²-h²+12h+4-4900=0

8h²+12h-4896=0

divide both sides by 4

2h²+3h-1224=0

factor 1224 into 2x2x2x3x3x17and then times 2 (from 2h²)

2x2x2x2x3x3x17=(2x2x2x2x3)(3x17) and 51-48=3, the coefficient of the middle term

2 and 24 must multiply together and 1 and 51 must also (24x51=1224)

(2h        )(h     )=0

(2h+51)(h-24)=0

disregard (2h+51) because it gives a negative answer

h-24=0

h=24 feet, the height of the tree

Answer 2

70 is the length for the base of the triangle.

 

2 + 3H is the distance to the tope of the tree (hypotenuse).

 

H is the height of the tree.

 

 

Using the Pythagorean theorem: 70² + H² = (2 + 3H)²

 

 

4900 + H² = 4 + 12H + 9H²

 

 

8H² + 12H - 4896 = 0

 

 

2H² + 3H - 1224 = 0

 

 

Using the quadratic formula,

 

H = -3/4 + (1/4)√[9 + 4(2)(1224)] = -3/4 + 99/4 = 96/4 = 24 feet